Int. J. Engng Ed. Vol. 20, No. 5, pp. 867±871, 2004
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# 2004 TEMPUS Publications.
The Use of Spreadsheets to Calculate the
Convolution Sum of Two Finite
Sequences*
ALI EL-HAJJ, KARIM Y. KABALAN
Electrical and Computer Engineering Department, American University of Beirut, P.O. Box 11-0236,
Beirut, Lebanon. Email: kabalan@aub.edu.lb
SHAHWAN KHOURY
Faculty of Engineering, Notre Dame University, Jounieh, Lebanon
This paper presents an educational spreadsheet programs for calculating the convolution sum of
two finite sequences. This convolution sum is accomplished using three different methods. After
each method is presented, the procedure for computing the convolution sum is outlined, and the
results are obtained for each of these cases. These explicit methods of calculation are proven to be
well suited for an educational environment as characterized by their flexibility and simplicity.
Moreover, when such software is used, they could make engineering studies more appealing to the
students.
constructing these tables and for providing an
invaluable insight into the nature of the problem.
As a presentation of the sensitivity concept of
the addressed problem, this paper illustrates the
three different methods used to construct these
sum tables and thus the convolution sums. The
first two methods [11] represent a classical
approach to construct the convolution tables
where the third method introduced in [12] uses a
table similar to the one constructed for the multiplication of two decimal numbers.
This spreadsheet approach presents a comprehensive procedure and an attractive educational
tool. This ideal process has the following characteristics:
INTRODUCTION
SPREADSHEETS are widely used for solving
engineering problems and are a well-recognized
engineering education tool providing the practical
ground for presenting theoretical concepts that
form the foundation of a subject. In this respect,
methods were developed to simulate logic
networks [1±2], the operation of an analog computer [3], and the function of microprocessor
systems [4]. Moreover, spreadsheets were also
used to analyze linear, nonlinear, and sampled
data control systems [5±6], and to study the
stability of linear and discrete control systems
[7±8]. Furthermore, some signal analysis exercises
using spreadsheets were considered in [9] which
include the calculation of the convolution integral
of two functions.
In this paper, the use of modern spreadsheet
programs [10] for calculating the convolution sum
of two finite sequences is considered. This discrete
convolution is parallel to that used for continuous
functions considered in [9]. It has applications in
many engineering areas such as discrete signal
analysis, discrete filtering, and communications.
The problem faced by beginner students in learning discrete convolution is covered in [11] where
the calculations require a number of iterations that
need effort and time to perform. To develop a
coherent knowledge of the subject, some useful
approaches for uninitiated students are presented
in [11±12], where structured tables of values of the
sequences were constructed. In this paper, it is
shown that spreadsheets are suitable tools for
1. It is of low cost since spreadsheets are widely
available.
2. It is possible to study the influence of changing
a value in any of the sequences on the resulting
convolution sum.
3. In Excel, the zooming option allows viewing a
large table on one screen and making the most
efficient use of the screen.
FIRST METHOD
This method uses the approach presented in
Example 6.3.4 of reference [11] to construct a
table. In this method, the basic definition of the
convolution of two finite causal sequences (the
sequence has zero value for negative values of n)
is used with the fact that the convolution of an
M-point sequence x[n] and an N-point sequence
h[n] is an (M ‡ N ÿ 1)-point sequence y[n]. In a
* Accepted 21 January 2004.
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A. El-Hajj et al.
mathematical form:
y‰nŠ ˆ
N‡Mÿ1
X
h‰kŠx‰n ÿ kŠ
1†
kˆ0
In this example, h[n] is a five-point causal
sequence and x[n] is a four-point causal sequence.
Thus, y[n] is an eight-point sequence defined from
n ˆ 0 to n ˆ 7. It can be easily deduced from
Equation (1), that (n ÿ k) takes all values
from ÿ k to 7 ÿ k as n takes values from 0 to 7.
As a result, y[n] can be considered as the product
of the vector h[k] and the vector x[n ÿ k] which is
obtained from x[k] using an appropriate transformation.
This transformation is constructed in the worksheet of Fig. 1 with the following steps:
1. The first row of the table, row 5, is filled with all
values of k used in evaluating the vectors
x[n ÿ k]. The smallest value is ÿ 3 used in
x[ÿ k] and the largest is 7 used in x[7 ÿ k].
2. The second and the third rows of the table are
used to input the given values of the sequences
h[n] and x[n].
3. The entries of the remaining rows of the table
are obtained by first reflecting x[k] about the
origin to form x[ÿ k], and then shifting the
resulting sequence to the right one step at a
time to form the remaining vectors.
4. The M column of the table, labeled n, indicates
the values of n for which y[n] will be calculated.
In this example, and as mentioned earlier, n
takes the values from 0 to 7.
5. The N column of the table, labeled y[n], is used
to store the results of the multiplication of the
entries in the rows corresponding to h[k] (range
B6:L6) and x[n ÿ k]. To perform the product of
these two vectors, the transpose ht[k] of h[k] is
calculated at range P6:P16 (using the spreadsheet command Paste Special), then y[0] is
obtained by writing at cell N8 the matrix
multiplication formula:
ˆ MMULT…B8:L8; $P$6:$P$16†
2†
The transpose of the sequence h[n] is obtained due
to the restriction in matrix manipulation in Excel.
Formula (1) is then copied to range N9:N15 to
obtain the required values of the sequence y[n].
SECOND METHOD
This method uses the approach discussed in
Example 6.3.5 of reference [11] to construct the
convolution table. In this method the convolution
sum y[n] of the two finite sequences h[n] and x[n] is
given by the formula:
y‰nŠ ˆ h‰0Šx‰nŠ ‡ h‰1Šx‰n ÿ 1Š ‡ : : :
‡h‰kŠx‰n ÿ kŠ ‡ : : : ‡ h‰mŠx‰n ÿ mŠ
3†
In Formula (3), m is the highest index for which
h[m] is different than zero. To present this procedure, the same example taken before is restated in
this section. In this example, y[n] takes values from
n ˆ 0 to n ˆ 7. Consequently, the values of the
sequences x[n], x[n ÿ 1], x[n ÿ 2], x[n ÿ 3], and
x[n ÿ 4] only are needed to compute y[n].
The procedure along with the computation
directions are conferred in Fig. 2 using the following steps:
1. A header row, row 5, is filled with all needed
values of n. This row also contains headers to
indicate that k and h[k] will be stored in the
corresponding columns to facilitate the multiplication procedure as will be explained later.
Fig. 1. Convolution table using the first method.
The Use of Spreadsheets to Calculate the Convolution Sum of Two Finite Sequences
869
Fig. 2. Convolution table using the second method.
2. The vectors x[n], x[n ÿ 1], . . . x[n ÿ m] are
stored in the next rows. First, the values of
the sequence x[n] are inputted in row 6. The
other rows, rows 7±10, are obtained by simple
shifting of the range containing x[n] . This one
place to the right shifting is done when copying
to the next row.
3. The next rows, rows 11±15, contain the terms
constituting the sum in Formula (3). The first
row contains the terms h[0]x[n], the second
contains h[1]x[n ÿ 1], and so on. To facilitate
obtaining these terms, each h[k] value is stored
at the same row as the corresponding x[n ÿ k] as
indicated earlier. The formula at cell B11 is
given by: ˆ ‡B6*$K6.
4. The use of mixed addressing (relative and
absolute addressing) allows obtaining the
other h[k]x[n ÿ k] terms by simply copying this
formula to the range B11:I15.
5. The convolution sum is next obtained by writing at cell B16 the formula: ˆ SUM(B11:B15)
This formula is copied to the range C16:I16. As an
alternative of writing these formulas, it is possible
to make use of the AutoSum Button found on the
Standard Toolbar and apply it to the range
B11:I15. Figure 2 shows similar results as in the
previous method and as expected.
THIRD METHOD
This method uses the approach discussed in [12]
to construct the table of Fig. 3. This procedure is
similar to the multiplication of two decimal
numbers which makes this method attractive,
easy to learn, and simple to implement.
To obtain this table, the following steps are
done:
Fig. 3. Convolution table using the third method.
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A. El-Hajj et al.
1. The two sequences are placed in two consecutive rows. In this case, a number of columns
equal to the number of elements of the sequence
x[n] should be skipped in order to accommodate the results of the multiplication. Figure 3
shows the placement of the two given sequences
right aligned at rows 6 and 7 after skipping 4
columns (columns A to D). To outline the
approach, a line is drawn under x[n] to graphically demonstrate the multiplication procedure.
2. The rightmost x[n] value is multiplied by the
vector h[n]. The result of the multiplication is
stored in the next row, row 8, as described in the
procedure. This is done by writing at cell I8 the
formula: ˆ ‡$I$7*I6. This formula is copied to
the range H8:E8.
3. Similarly, the next x[n] value (from the right) is
multiplied by the vector h[n]. The result of this
operation is stored in the next row. The procedure
is continued for all elements of x[n] in a fashion
similar to a pencil and paper multiplication
calculation, except carries are not performed
out of a column [12]. Shifting one place to the
left is done each time a next row is calculated. A
line is finally drawn under the last calculated
row to indicate the end of the procedure.
4. To obtain the convolution sum y[n], the vectors
h[n]x[3], h[n]x[2], h[n]x[1], and h[n]x[0] are
added together. The result is computed and
stored in row 12. This is done by writing in
cell B12 the formula: ˆ SUM(B8:B11).
This formula is copied to the range C12:I12. As an
alternative of writing these formulas, it is possible
to make use of the AutoSum Button that is found
on the Standard Toolbar and apply it to the range
B8:I11. Finally, the corresponding values of n are
filled in in row 13. Similar results are obtained as in
the previous methods.
CONCLUSION
The use of modern spreadsheet programs for
calculating the convolution sum of two finite
causal sequences was considered. Three different
methods were presented and examples of evaluation procedures are given. They are based on using
formulas to construct a table showing different
steps in calculating the convolution sum. These
methods facilitate a simple, flexible, and userfriendly tool suitable for educational environments
and spreadsheet users. It is possible to study the
influence of changing a value in any of the
sequences on the resulting convolution sum by
changing this value on the worksheet which
causes the worksheet to recalculate providing the
new y[n] values. It requires less time/effort to study
and implement than manual calculations and/or
conventional programming. The same methods,
which are presented for two causal systems, can
be extended to treat non-causal systems.
REFERENCES
1. A. El-Hajj and K. Y. Kabalan, A spreadsheet simulation of logic networks, IEEE Trans. Educ.,
E-34, 1991, pp. 43±46.
2. A. El-Hajj, K. Y. Kabalan and M. Yehia, Simulation of a class of integrated circuits using
spreadsheets, IEE Proceedings-G, 139(5) 1992, pp. 607±610.
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spreadsheets, Simulation Journal of the Society for Computer Simulation, 68(2) February 1997,
pp. 101±106.
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systems, J. Computer Applications in Engineering Education, 3(3), 1995, pp. 205±211.
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IEEE Trans. Educ., 38(4) 1995, pp. 317±320.
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using spreadsheets, IEE Proceedings A, 143(1) 1996, pp. 52±56.
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systems, Int. J. Appl. Engng. Educ., 6(4) 1990, pp. 437±442.
8. K. Y. Kabalan, H. Diab, A. El-Hajj and H. Tabbarah, Stability analysis and minimum phase
testing of casual systems using spreadsheets, Int. J. Eng. Educ., 12(5) 1996, pp. 382±388.
9. B. J. Stanton, M. J. Drozdowski and T. S. Ducan, Using spreadsheets in student exercises for signal
and linear system analysis, IEEE Trans. Educ., 36(1) 1993, pp. 62±67.
10. Microsoft Excel User's Guide, Version 5.0, Microsoft Corporation (1994).
11. S. S. Soliman, and M. D. Srinath, Continuous and Discrete Signals and Systems, Prentice-Hall
(1990).
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Ali El-Hajj was born in Aramta, Lebanon. He received the Lisense degree in Physics from
the Lebanese University, Lebanon in 1979, the degree of `Ingenieur' from L'Ecole Superieure d'Electricite, France in 1981, and the `Docteur Ingenieur' degree from the University
of Rennes I, France in 1983. From 1983 to 1987, he was with the Electrical Engineering
Department at the Lebanese University. In 1987, he joined the American University of
The Use of Spreadsheets to Calculate the Convolution Sum of Two Finite Sequences
Beirut where he is currently Professor of Electrical and Computer Engineering. His research
interests are numerical solution of electromagnetic field problems and engineering
education.
Karim Y. Kabalan was born in Jbeil, Lebanon. He received the BS degree in Physics from
the Lebanese University in 1979, and the MS and Ph.D. degrees in Electrical and Computer
Engineering from Syracuse University, in 1983 and 1985, respectively. During the 1986 Fall
semester, he was a visiting assistant professor of Electrical and Computer Engineering at
Syracuse University. Currently, he is a Professor of Electrical and Computer Engineering
with the Electrical and Computer Engineering Department, Faculty of Engineering and
Architecture, American University of Beirut. His research interests are numerical solution
of electromagnetic field problems and software development.
Shahwan V. Khoury: was born in Ghalboun, Lebanon. He received his BE. degree in
Electrical Engineering from Youngstown State University in 1960. He also received his MS
and Ph.D. degrees in Electrical Engineering from Carnegie Institute of Technology in 1961,
and 1964, respectively. He was a member of the Technical Staff (AT&T), Technical
Director at Projects Studies and Implementation Co. (PSAI), Geneva and San Francisco,
Manager of Electrical and Mechanical Works Contracts Division of Engineering and
General Contracting, Ets. F.A. Kettaneh, Beirut, Lebanon, General Manager, Debbas
Enterprise Beirut, Lebanon, and Consultant, SDI, London. Currently, he is the Dean and
Professor of Engineering, Faculty of Engineering, Notre Dame University, Jounieh,
Lebanon. His research interests are in numerical solution of electromagnetic field problems
and software development.
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